1975 AMC 12 Problem 23

Attempt Problem 23 of the 1975 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1975 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

In the adjoining figure

AB

and

BC

are adjacent sides of square

ABCD;

M

is the midpoint of

AB;

N

is the midpoint of

BC;

and

AN

and

CM

intersect at

O.

The ratio of the area of

AOCD

to the area of

ABCD

\frac56

\frac34

\frac23

\frac{\sqrt3}{2}

\frac{\sqrt3-1}{2}

In triangle

ABC,

both

AN

and

CM

are medians

Use coordinates for the square and locate the centroid

O

Take

A=(0,0),B=(1,0),

C=(1,1),D=(0,1).

Since

O

is the centroid of triangle

ABC,

O=(\frac{2}{3},\frac{1}{3}).

Triangles

AOB

and

COB

each have area

\frac{1}{6}.

Removing them from the unit square leaves

[AOCD]=1-\frac16-\frac16=\frac23.

Therefore, the correct answer is C.

Problem 23 in other years